Center (group theory)

In abstract algebra, the center of a group, , is the set of elements that commute with every element of . It is denoted , from German '' Zentrum,'' meaning ''center''. In set-builder notation,

:.

The center is a normal subgroup, . As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, , is isomorphic to the inner automorphism group, .

A group is abelian if and only if . At the other extreme, a group is said to be centerless if is trivial; i.e., consists only of the identity element.

The elements of the center are sometimes called central.

As a subgroup

The center of ''G'' is always a subgroup of . In particular:
# contains the identity element of , because it commutes with every element of , by definition: , where is the identity;
# If and are in , then so is , by associativity: for each ; i.e., is closed;
# If is in , then so is as, for all in , commutes with : .

Furthermore, the center of is always a normal subgroup of . Since all elements of commute, it is closed under conjugation.

Note that a homomorphism between groups generally does not restrict to a homomorphism between their centers. Although commutes with , unless is surjective need not commute with all of and therefore need not be a subset of . Put another way, there is no "center" functor between categories Grp and Ab. Even though we can map objects, we cannot map arrows.

Conjugacy classes and centralizers

By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., .

The center is also the intersection of all the centralizers of each element of . As centralizers are subgroups, this again shows that the center is a subgroup.

Conjugation

Consider the map, , from to the automorphism group of defined by , where is the automorphism of defined by
:.

The function, is a group homomorphism, and its kernel is precisely the center of , and its image is called the inner automorphism group of , denoted . By the first isomorphism theorem we get,
:.

The cokernel of this map is the group of outer automorphisms, and these form the exact sequence
:.

Examples

* The center of an abelian group, , is all of .
* The center of the Heisenberg group, , is the set of matrices of the form: $\begin 1 & 0 & z\\ 0 & 1 & 0\\ 0 & 0 & 1 \end$
* The center of a nonabelian simple group is trivial.
* The center of the dihedral group, , is trivial for odd . For even , the center consists of the identity element together with the 180° rotation of the polygon.
* The center of the quaternion group, , is .
* The center of the symmetric group, , is trivial for .
* The center of the alternating group, , is trivial for .
* The center of the general linear group over a field , , is the collection of scalar matrices, .
* The center of the orthogonal group, is .
* The center of the special orthogonal group, is the whole group when , and otherwise when ''n'' is even, and trivial when ''n'' is odd.
* The center of the unitary group, $U(n)$ is $\left\$.
* The center of the special unitary group, $\operatorname(n)$ is $\left\lbrace e^ \cdot I_n \mid \theta = \frac, k = 0, 1, \dots, n-1 \right\rbrace$.
* The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.
* Using the class equation, one can prove that the center of any non-trivial finite p-group is non-trivial.
* If the quotient group is cyclic, is abelian (and hence , so is trivial).
* The center of the megaminx group is a cyclic group of order 2, and the center of the kilominx group is trivial.

Higher centers

Quotienting out by the center of a group yields a sequence of groups called the upper central series:

:

The kernel of the map is the th center of (second center, third center, etc.) and is denoted . Concretely, the ()-st center are the terms that commute with all elements up to an element of the th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter.This union will include transfinite terms if the UCS does not stabilize at a finite stage.

The ascending chain of subgroups
:
stabilizes at ''i'' (equivalently, ) if and only if is centerless.

Examples

* For a centerless group, all higher centers are zero, which is the case of stabilization.
* By Grün's lemma, the quotient of a perfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at .

See also

*Center (algebra)
*Center (ring theory)
* Centralizer and normalizer
* Conjugacy class

Notes

References

*

External links

* {{springer, title=Centre of a group, id=p/c021250

Group theory
Functional subgroups